Localized Pressure and Equilibrium States

Abstract

We introduce the notion of localized topological pressure for continuous maps on compact metric spaces. The localized pressure of a continuous potential $$\varphi $$ is computed by considering only those $$(n,\epsilon )$$ -separated sets whose statistical sums with respect to an m-dimensional potential $$\Phi $$ are “close” to a given value $$w\in {\mathbb R}^m$$ . We then establish for several classes of systems and potentials $$\varphi $$ and $$\Phi $$ a local version of the variational principle. We also construct examples showing that the assumptions in the localized variational principle are fairly sharp. Next, we study localized equilibrium states and show that even in the case of subshifts of finite type and Holder continuous potentials, there are several new phenomena that do not occur in the theory of classical equilibrium states. In particular, we construct an example with infinitely many ergodic localized equilibrium states. We also show that for systems with strong thermodynamic properties and w in the interior of the rotation set of $$\Phi $$ there is at least one and at most finitely many localized equilibrium states.